Two-Dimensional Elastodynamic Simulations

SUPPORT-TUP

4.1 Two-Dimensional Elastodynamic Simulations

The simulations of the dynamic experiments were first attempted under the simplifying assumption that the specimens could be considered to be under essen- tially plane-stress conditions. Before delving into the simulations of the experiments one issue needs to be addressed. Since no special singularity elements were used in the finite-element analyses, it is essential that the discretization used must be such as to capture the expected singular crack-tip fields adequately. To this effect, preliminary two-dimensional elastostatic analyses of the three-point bend specimen were performed. Based on the results of these, the mesh discretization shown in Figure 4.2 consisting of 396 isoparametric linear quadrilateral elements ( 425 nodes) was found to be adequate. This mesh has a focussed region around the crack-tip of about one (actual) specimen thickness which is divided into 18 sectors and 10 concentric rings of elements as shown in Figure 4.2. For the two-dimensional elas- tostatic case, the solution given by Williams (1957) indicates that the crack-tip stress-fields must be square-root singular. As a check of whether the crack-tip sin- gularity is captured by this discretization, a logarithmic plot of stress component

Psupport

X2 X2

H

I I I • I (

Ptup • ., X1

X3

Figure 4.2: Finite-element discretization; on the right is ::;hown the mesh gradation through half the ::;pccimen thickne::;s u::;e<l in the three-dimensional simulations.

I (JI 0 ) I

-57-

o-11 along the fJ

=

45° line is shown in Figure 4.3 as a function of the logarithm of radial distance to the crack-tip. Also shown are the corresponding asymptotic values with the magnitude of the stress-intensity factor obtained from the J-integral ( computed using the domain integral formulation discussed in Appendix Al). The mesh discretization is seen to be adequate to model the square-root singular field in the elastostatic case and is thus expected to be suitable for the dynamic problem as well.

For the two-dimensional elastodynamic simulations, the loads as obtained from the experimental tup records were applied as the boundary conditions. That is, the impact-tup load history was applied to the node corresponding to the impact-tup and the support-tup load history was applied to the associated node as shown in Figure 4.2. From symmetry conditions, the uncracked ligament was constrained to move only along the x ₁ -direction. The rest of the boundary was left free of traction. An implicit Newmark predictor-corrector time integration scheme ( see Appendix A2) was used for its virtue of unconditional stability which would allow for relatively large time steps and the attendant loss of high frequency information was deemed acceptable since it is not the intent here to monitor discrete stress waves in the body.

The (virtual) energy-release rate for a dynamically-loaded stationary crack is given by (see Appendix Al)

J

=

_{r--o Jr} ^lim

f ((u +

^T)n1 - o-iin1 _8x

aui) ^dr

1

( 4.1) where U is the strain-energy density, T is the kinetic energy density, O"ij is the stress-tensor, 1£ is the displacement vector, and !l is the unit outward normal to the contour of integration

r.

Here,

r

-+ 0 symbolically indicates that the integration contour must be shrunk on to the crack-tip. In the simulations, the time history of this integral was computed using the equivalent domain integral form as explained

4 - - - -

2

~ o

;I __

~~

...

~

'--"' -2

bO 0

...

-4

0

=

45°

-- --

--- finite-element ---- asymptotic

-6 ___ __,, _ _ _ _ _ _._ _ _ _ _ _ _ ,.__ _ ___,, _ _ _ _ _ __.

-6.0 -5.0 -4.0 -3.0 -2.0 -1.0 .0 1.0 2.0

log(r/h)

Figure 4.3: Plot of log

(Ki/~)

^{versus log(r / h)} for the two-dimensional elastostatic case.

I

C.H

CX) I

-59-

in Appendix Al. The dynamic stress-intensity factor was then computed through the relation between

Kf

^and J in plane-stress,

(4.2) where Eis the Young's modulus of the material. Figures 4.4a,b show the experimen- tally obtained dynamic stress-intensity factor history in comparison with that from the numerical simulations for specimens ( v3s) and ( a - 4). Here

Kf

₁ and

Kf

₂ refer to the experimentally measured values corresponding to the two object plane dis- tances zo1 and zo2 and Kj is that computed from the dynamic simulations ( through the J-integral). It is seen that in both cases Kj has the same general trend as ^J{Il and K12 except that the experimental values are sometimes substantially lower, while at other times equal to or higher than the simulated values. This discrep- ancy is attributable to two sources. First, there are uncertainties associated with the simulations in terms of how accurately the tup records provide the boundary tractions actually experienced by the specimens. Secondly, and more importantly, there is the possibility that the experimental values might not have been obtained from a region of Kf-dominance. This is in fact foreshadowed by the discrepancy between the two experimental records themselves.

In the above, it has been implicitly assumed that the asymptotic Kf-field has validity for this (two-dimensional) geometry and dynamic loading condition. How- ever, as pointed out in the introductory chapter, the existence of a stress-intensity factor field around a dynamically-loaded stationary crack in a finite geometry has by no means been universally established. It is thus necessary to check whether a square-root singular asymptotic field is appropriate here. To this end, a logarithmic plot of stress component o-₂₂ along the 0

=

5° line versus the logarithm of radial distance from the crack-tip is shown in Figure 4.5 for two representative times in the simulation. Comparison with the corresponding curves for the asymptotic field

250 ..---r---r---r---.---.

200

-

--!; 150

- ~

_~

₁₀₀

50

- - finite-element (I<j) ---· experiment

•2

01 = 3.84m

(JCf1)

~

z

O 2 = 2. 46m

(1(1

_{2 )}

I I I

I I~

/

I I

;"-I

,.._,,i /

I ,,__ I I / "

I I

I /:,.

I /

,...J I I

/ /

,.--A-.___ "\_J- I

/ /

----,

^/"./

/

'--,

^{/ ~}

/ -'t{'

/ ,,,----✓

%

_I

I

,K / ..-_,,,.---✓

(/

,.Ir'

. 'l / / / I 1/ / 1/ / / / / /

/ _ / /

0 ...

=---...L.---'---'---.,___ _ _ __,

0

100 200 300 400 500